Let \(f:X\to Y\) be a map of ringed spaces and \(M\) and \(N\) \(\mathcal O_Y\) modules. Under what conditions to we have \[f^*\mathscr{H}om_{\mathcal O_Y}(M, N) \cong \mathscr{H}om_{\mathcal O_X}(f^*M, f^*N)?\]
For context, I’m looking at the conormal sequence for a closed embedding \(i:X\to V\) of a n.c. variety into a smooth ambient space \(V\): \[0 \to I/I^2 \to \Omega^1_{V}|_X \to \Omega^1_X\to 0\] When I take the dual I get \[0 \to \Theta_X\to \mathscr{H}om(\Omega^1_V|_X, \mathcal O_X) \to N_{V|X} \] and I’d like to know that the middle term is in fact \(\Theta_V|_X\), i.e. that it’s the dual of the tangent bundle on \(V\) pulled back to \(\mathcal O_X\). For this I need \[\mathscr Hom_{\mathcal O_X}(i^*\Omega_{V}|_{X}, \mathcal O_X) = i^*\mathscr Hom_{\mathcal O_V}(\Omega^1_V|_X, \mathcal O_V) = \Theta_V|_X.\]