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?

  • $\begingroup$ Do you have a question? $\endgroup$ Commented Oct 9, 2019 at 5:44
  • $\begingroup$ Oh, I was asking whether this is correct. Sorry. $\endgroup$
    – Bryan Shih
    Commented Oct 9, 2019 at 5:45
  • $\begingroup$ @KevinCarlson, if you don't mind, I would be grateful if you could give a look at my question on luries HTT. I am still confused how is there an induced map. $\endgroup$
    – Bryan Shih
    Commented Oct 9, 2019 at 5:47
  • 1
    $\begingroup$ Yes, this is correct. $\endgroup$ Commented Oct 9, 2019 at 5:49

1 Answer 1


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.


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