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Using the IllustrisTNG simulations, we investigate the connection betweengalaxy morphology and star formation in central galaxies with stellar masses inthe range $10^9-10^{11.5}~\mathrm{M}_{\odot}$. We quantify galaxy morphology bya kinematical decomposition of the stellar component into a spheroidal and adisc component (spheroid-to-total ratio, S/T) and by the concentration of thestellar mass density profile ($C_{82}$). S/T is correlated with stellar massand star-formation activity, while $C_{82}$ correlates only with stellar mass.Overall, we find good agreement with observational estimates for both S/T and$C_{82}$. Low and high mass galaxies are dominated by random stellar motion,while only intermediate-mass galaxies($M_{\star}\approx10^{10}-10^{10.5}~\mathrm{M}_{\odot}$) are dominated byordered rotation. Whereas higher-mass galaxies are typical spheroids with highconcentrations, lower-mass galaxies have low concentration, pointing todifferent formation channels. Although we find a correlation between S/T andstar-formation activity, in the TNG model galaxies do not necessarily changetheir morphology when they transition through the green valley or when theycease their star formation, this depending on galaxy stellar mass andmorphological estimator. Instead, the morphology (S/T and $C_{82}$) isgenerally set during the star-forming phase of galaxies. The apparentcorrelation between S/T and star formation arises because earlier-forminggalaxies had, on average, a higher S/T at a given stellar mass. Furthermore, weshow that mergers drive in-situ bulge formation in intermediate-mass galaxiesand are responsible for the recent spheroidal mass assembly in the massivegalaxies with $M_{\star}>10^{11}~\mathrm{M}_{\odot}$. In particular, thesemassive galaxies assemble about half of the spheroidal mass while star-formingand the other half through mergers while quiescent.

Morphology and star formation in IllustrisTNG: the build-up of spheroids and discs