The paths of these two stars have intersected on several occasions due to their association with the luxury brand Bulgari
The tourists were 152 for six at stumps in their second innings, still needing 89 runs to avoid an innings defeat in India's pink ball Test debut in Kolkata.
Kohli made 136 as India, who lead the two-match series 1-0, declared their innings on 347 for nine and then Ishant rattled the Bangladesh batting.
Ishant struck early to leave Bangladesh tottering at 13 for four before Mushfiqur Rahim made an unbeaten 59 to take the five-day match to day three.
Mushfiqur and Mahmudullah, who retired hurt on 39 with what seemed a hamstring strain, put on 69 runs for the fifth wicket to offer some resistence against a persistent Indian pace attack.
Mushfiqur stayed on to complete his second half-century of the series with concussion substitute Taijul Hasan losing his wicket for 11 to paceman Umesh Yadav and umpires called for end of play.
Mushfiqur survived two leg before wicket decisions that went against him but he successfully reviewed both the calls in his favour.
Ishant trapped Shadman Islam lbw for nought and then claimed skipper Mominul Haque for a second zero in the same match.
The pace spearhead, who claimed five wickets in the first innings, bowled at a lively pace as one of his rising deliveries hit Mohammad Mithun on the helmet but the batsman went on to complete his innings.
Bangladesh were forced to take two concussion substitutes in Mehidy Hasan and Taijul for Liton Das and Nayeem Hasan who were hit on the helmet by Shami on day one.
Earlier Kohli aced the pink ball Test as he completed his 27th century before falling to a stunning catch by Taijul.
Left-arm spinner Taijul made it count with a wicket of Ajinkya Rahane for 51 and then took a diving catch at fine leg to send back Kohli off paceman Ebadat Hossain.
India lost a few quick wickets, but Wriddhiman Saha and Shami knocked a few boundaries and a six by the last man in pulverised the Bangladesh bowlers.
Pacemen Al-Amin Hossain and Ebadat took three wickets each.
Kohli stood out with his prolific scoring as he put on crucial partnerships including a 99-run stand with overnight partner Rahane.
The star batsman smashed fast bowler Abu Jayed for four straight boundaries in one over before finishing with 18 fours in his 194-ball stay.Scoreboard
Bangladesh 1st innings 106 (Shadman Islam 29; I. Sharma 5-22)
India 1st innings (overnight 174-3, Kohli 59, Rahane 23)
M. Agarwal c Mehidy b Al-Amin 14
R. Sharma lbw b Ebadat 21
C. Pujara b Ebadat 55
V. Kohli c Taijul b Ebadat 136
A. Rahane c Ebadat b Taijul 51
R. Jadeja b Abu 12
W. Saha not out 17
R. Ashwin lbw b Al-Amin 9
U. Yadav c Shadman b Abu 0
I. Sharma lbw b Al-Amin 0
M. Shami not out 10
Extras (b12, lb2, w8) 22
Total (9 wkts dec, 89.4 overs) 347
Fall of wickets: 1-26, 2-43, 3-137, 4-236, 5-289, 6-308, 7-329, 8-330, 9-331
Bowling: Al-Amin 22.4-3-85-3 (w5), Abu 21-6-77-2, Ebadat 21-3-91-3 (w3), Taijul 25-2-80-1
Bangladesh 2nd innings
S Islam lbw b Ishant 0
I Kayes c Kohli b Ishant 5
M Haque c Saha b Ishant 0
M Mithun c Shami b Yadav 6
M Rahim not out 59
Mahmudullah retired hurt 39
M Hasan c Kohli b Ishant 15
T Islam c Rahane b Yadav 11
Extras (b8, lb4, w5) 17
Total (6 wickets, 32.3 overs) 152
To bat: Abu Jayed, Al-Amin Hossain, Ebadat Hossain
Fall of wickets: 1-0, 2-2, 3-9, 4-13, 5-133
Bowling: Ishant 9-1-39-4, Yadav 10.3-0-40-2, Shami 8-0-42-0 (w5), Ashwin 5-0-19-0
The paths of these two stars have intersected on several occasions due to their association with the luxury brand Bulgari
The students' demands range from a ceasefire in Israel's war with Hamas to calls for universities to stop investing in Israeli enterprises
Despite Kohli's flowing form ahead of the T20 World Cup in June, pundits have raised concerns over his strike-rate
Saudi Arabia has begun to explore extracting lithium from sea water
Speaking after meeting Lebanese leaders, French foreign minister says there had been "a lot of progress" over the French proposals
The three-shot winner from Japan says he is aiming to become a member of the PGA Tour in the future
Big-hitting Nienaber falters at the finish to settle for secondwhile former Abu Dhabi resident David Horsey takes the third spot at Saadiyat Beach Golf Club
The UAE is India's second-largest export destination after the US