I'm reading Hartshorne's *Deformation Theory* book. In exercise 5.9(b), he claims:

Let $C$ be a reduced projective locally complete intersection curve embedded inside a smooth surface $X$ (over some alg. closed field $k$), as: $$i : C\hookrightarrow X$$ Tensoring the exact sequence $$0\rightarrow T_{C/k}\rightarrow i^*T_{X/k}\rightarrow N_{C/X}\rightarrow T^1_{C/k}\rightarrow 0$$ with $\omega_{X/k}$ (the dualizing sheaf of $X$, ie $\omega_{X/k} = \det\Omega_{X/k}$), we get a sequence $$0\rightarrow T_{C/k}\otimes\omega_{X/k}\rightarrow i^*\Omega_{X/k}\rightarrow \omega_{C/k}\rightarrow T^1_{C/k}\otimes\omega_{X/k}\rightarrow 0$$ here $T = \Omega^\vee$ denotes the relative tangent sheaf, and $N$ the relative normal sheaf. The extremal terms of the sequence obviously are obtained by tensoring with $\omega_{X/k}$. The third term is $\omega_{C/k}$ by essentially the definition of the dualizing sheaf of an LCI scheme, so my question concerns the second term -

Why is $i^*T_{X/k}\otimes\omega_{X/k}\cong i^*\Omega_{X/k}$? Here, I assume the left hand side should be viewed as $$i^*T_{X/k}\otimes\omega_{X/k} := T_{X/k}\otimes_{\mathcal{O}_X}\omega_{X/k}\otimes_{\mathcal{O}_X}\mathcal{O}_C = i^*\left(T_{X/k}\otimes\omega_{X/k}\right)$$

Exercise II.5.16 (b). Let me write out some details and post it as an answer. $\endgroup$ – R. van Dobben de Bruyn Mar 17 '17 at 21:10