# Internal hom: products and coproducts

It seems to me that

Assuming the all smalllimits and colimits exist: Internal hom for a closed symmetric monoidal category satisfies: $$[\bigsqcup C_i, X] \cong \prod_i [C_i, X]$$ where $$(-)\otimes X: V \rightarrow V$$, have right adjoint $$[X,-]:V \rightarrow V$$.

My proof is by showing that they represent the same object in $$V$$. Is this isomorphism true?

• Do you have a question? Commented Oct 9, 2019 at 5:44
• Oh, I was asking whether this is correct. Sorry. Commented Oct 9, 2019 at 5:45
• @KevinCarlson, if you don't mind, I would be grateful if you could give a look at my question on 5.4.1.1 luries HTT. I am still confused how is there an induced map. Commented Oct 9, 2019 at 5:47
• Yes, this is correct. Commented Oct 9, 2019 at 5:49

You should show us your proof, but the standard proof carries out essentially like the proof that left adjoints preserve colimits and right adjoints preserve limits. $$[\_, X]$$ actually turns all colimits into limits, not just coproducts into products ($$[\_, X]: V^{op} \to V$$ preserves limits).
For objects $$X$$ and $$Y$$ in $$V$$ and a diagram $$F: J \to V$$,
\begin{aligned} C(X, [ \mathrm{colim}_{j \in J} F(j), Y ] ) & \simeq C(X \otimes \mathrm{colim}_{j \in J} F(j), Y ) \\ & \simeq C(\mathrm{colim}_{j \in J} F(j) \otimes X, Y ) \\ & \simeq C(\mathrm{colim}_{j \in J} F(j), [X, Y] ) \\ & \simeq \lim_{j \in J} C(F(j), [X, Y] ) \\ & \simeq \lim_{j \in J} C(F(j) \otimes X, Y ) \\ & \simeq \lim_{j \in J} C(X \otimes F(j), Y ) \\ & \simeq \lim_{j \in J} C(X, [F(j), Y] ) \\ & \simeq C(X, \lim_{j \in J} [F(j), Y] ) \\ \end{aligned}
Since this is natural in $$X$$, $$[ \mathrm{colim}_{j \in J} F(j), Y ] \simeq \lim_{j \in J} [F(j), Y]$$ by the Yoneda lemma.