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მამუკა ჯიბლაძე
მამუკა ჯიბლაძე
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Ok, this follows trivially from the construction of limits in ${\bf Cat}$.

Let $I\ $ be a small category, let $X\colon I\to{\bf Cat}\ $ denote a functor, and let $Y=\text{colim\}_{i\in I}X_i\ $ be its colimit with inclusion maps $q_i\colon X_i\to Y$. Let $d,e\colon Y\to {\bf Set}\ $$Y=\text{colim}_{i\in I}X_i\ $ be its colimit with inclusion maps $q_i\colon X_i\to Y$. Let $d,e\colon Y\to {\bf Set}\ $ be any functors. The question posed here reduces to the question of whether the following function is an isomorphism: $$Hom_{Y-{\bf Set}}(d,e)\to \lim_{i\in I}\ Hom_{X_i-{\bf Set}}(q_i^{\ast}d,q_i^{\ast}e)$$

It indeed is an isomorphism because the category $Y-{\bf Set}\ $ of functors $Y\to{\bf Set}\ $ is the limit of the categories $X_i-{\bf Set}\ $, and we know that the objects and hom-sets of a limit in ${\bf Cat}$ are computed pointwise.

We now prove that the above observation implies that the functor $K$ above preserves limits. It suffices to fix each variable in the domain of $K$, because a limit in a product of categories is the product of the limits. Since $g_{\ast}$ and $f^{\ast}$ are right adjoints, $K$ is continuous in the second variable. It suffices to show that if $\delta\colon C\to{\bf Set}\ $ is any functor and $(Y,F,G)=\text{colim\}_{i\in I}(X_i,F_i,G_i)\ $ in ${\bf Cat}/(C\times D)$ then the map $$G_{\ast}F^{\ast}\delta\to\lim_{i\in I}(G_i)_{\ast}(F_i)^{\ast}\delta$$ is an isomorphism of $D$$(Y,F,G)=\text{colim}_{i\in I}(X_i,F_i,G_i)\ $ in ${\bf Cat}/(C\times D)$ then the map $$G_{\ast}F^{\ast}\delta\to\lim_{i\in I}(G_i)_{\ast}(F_i)^{\ast}\delta$$ is an isomorphism of $D$-sets.

Let $\epsilon\colon D\to{\bf Set}\ $ be a $D$-set. Using the Yoneda imbedding and the $(G^{\ast},G_{\ast})\ $ adjunction, it suffices to show that the function $$Hom_{Y-{\bf Set}}(F^{\ast}\delta,G^{\ast}\epsilon)\to\lim_{i\in I}Hom_{X_i-{\bf Set}}(F_i^{\ast}\delta,G_i^{\ast}\epsilon)$$ is a bijection, which was the observation above.

Ok, this follows trivially from the construction of limits in ${\bf Cat}$.

Let $I\ $ be a small category, let $X\colon I\to{\bf Cat}\ $ denote a functor, and let $Y=\text{colim\}_{i\in I}X_i\ $ be its colimit with inclusion maps $q_i\colon X_i\to Y$. Let $d,e\colon Y\to {\bf Set}\ $ be any functors. The question posed here reduces to the question of whether the following function is an isomorphism: $$Hom_{Y-{\bf Set}}(d,e)\to \lim_{i\in I}\ Hom_{X_i-{\bf Set}}(q_i^{\ast}d,q_i^{\ast}e)$$

It indeed is an isomorphism because the category $Y-{\bf Set}\ $ of functors $Y\to{\bf Set}\ $ is the limit of the categories $X_i-{\bf Set}\ $, and we know that the objects and hom-sets of a limit in ${\bf Cat}$ are computed pointwise.

We now prove that the above observation implies that the functor $K$ above preserves limits. It suffices to fix each variable in the domain of $K$, because a limit in a product of categories is the product of the limits. Since $g_{\ast}$ and $f^{\ast}$ are right adjoints, $K$ is continuous in the second variable. It suffices to show that if $\delta\colon C\to{\bf Set}\ $ is any functor and $(Y,F,G)=\text{colim\}_{i\in I}(X_i,F_i,G_i)\ $ in ${\bf Cat}/(C\times D)$ then the map $$G_{\ast}F^{\ast}\delta\to\lim_{i\in I}(G_i)_{\ast}(F_i)^{\ast}\delta$$ is an isomorphism of $D$-sets.

Let $\epsilon\colon D\to{\bf Set}\ $ be a $D$-set. Using the Yoneda imbedding and the $(G^{\ast},G_{\ast})\ $ adjunction, it suffices to show that the function $$Hom_{Y-{\bf Set}}(F^{\ast}\delta,G^{\ast}\epsilon)\to\lim_{i\in I}Hom_{X_i-{\bf Set}}(F_i^{\ast}\delta,G_i^{\ast}\epsilon)$$ is a bijection, which was the observation above.

Ok, this follows trivially from the construction of limits in ${\bf Cat}$.

Let $I\ $ be a small category, let $X\colon I\to{\bf Cat}\ $ denote a functor, and let $Y=\text{colim}_{i\in I}X_i\ $ be its colimit with inclusion maps $q_i\colon X_i\to Y$. Let $d,e\colon Y\to {\bf Set}\ $ be any functors. The question posed here reduces to the question of whether the following function is an isomorphism: $$Hom_{Y-{\bf Set}}(d,e)\to \lim_{i\in I}\ Hom_{X_i-{\bf Set}}(q_i^{\ast}d,q_i^{\ast}e)$$

It indeed is an isomorphism because the category $Y-{\bf Set}\ $ of functors $Y\to{\bf Set}\ $ is the limit of the categories $X_i-{\bf Set}\ $, and we know that the objects and hom-sets of a limit in ${\bf Cat}$ are computed pointwise.

We now prove that the above observation implies that the functor $K$ above preserves limits. It suffices to fix each variable in the domain of $K$, because a limit in a product of categories is the product of the limits. Since $g_{\ast}$ and $f^{\ast}$ are right adjoints, $K$ is continuous in the second variable. It suffices to show that if $\delta\colon C\to{\bf Set}\ $ is any functor and $(Y,F,G)=\text{colim}_{i\in I}(X_i,F_i,G_i)\ $ in ${\bf Cat}/(C\times D)$ then the map $$G_{\ast}F^{\ast}\delta\to\lim_{i\in I}(G_i)_{\ast}(F_i)^{\ast}\delta$$ is an isomorphism of $D$-sets.

Let $\epsilon\colon D\to{\bf Set}\ $ be a $D$-set. Using the Yoneda imbedding and the $(G^{\ast},G_{\ast})\ $ adjunction, it suffices to show that the function $$Hom_{Y-{\bf Set}}(F^{\ast}\delta,G^{\ast}\epsilon)\to\lim_{i\in I}Hom_{X_i-{\bf Set}}(F_i^{\ast}\delta,G_i^{\ast}\epsilon)$$ is a bijection, which was the observation above.

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David Spivak
David Spivak
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Ok, this follows trivially from the construction of limits in ${\bf Cat}$.

Let $I\ $ be a small category, let $X\colon I\to{\bf Cat}\ $ denote a functor, and let $Y=\text{colim\}_{i\in I}X_i\ $ be its colimit with inclusion maps $q_i\colon X_i\to Y$. Let $d,e\colon Y\to {\bf Set}\ $ be any functors. The question posed here reduces to the question of whether the following function is an isomorphism: $$Hom_{Y-{\bf Set}}(d,e)\to \lim_{i\in I}\ Hom_{X_i-{\bf Set}}(q_i^{\ast}d,q_i^{\ast}e)$$

It indeed is an isomorphism because the category $Y-{\bf Set}\ $ of functors $Y\to{\bf Set}\ $ is the limit of the categories $X_i-{\bf Set}\ $, and we know that the objects and hom-sets of a limit in ${\bf Cat}$ are computed pointwise.

We now prove that the above observation implies that the functor $K$ above preserves limits. It suffices to fix each variable in the domain of $K$, because a limit in a product of categories is the product of the limits. Since $g_{\ast}$ and $f^{\ast}$ are right adjoints, $K$ is continuous in the second variable. It suffices to show that if $\delta\colon C\to{\bf Set}\ $ is any functor and $(Y,F,G)=\text{colim\}_{i\in I}(X_i,F_i,G_i)\ $ in ${\bf Cat}/(C\times D)$ then the map $$G_{\ast}F^{\ast}\delta\to\lim_{i\in I}(G_i)_{\ast}(F_i)^{\ast}\delta$$ is an isomorphism of $D$-sets.

Let $\epsilon\colon D\to{\bf Set}\ $ be a $D$-set. Using the Yoneda imbedding and the $(G^{\ast},G_{\ast})\ $ adjunction, it suffices to show that the function $$Hom_{Y-{\bf Set}}(F^{\ast}\delta,G^{\ast}\epsilon)\to\lim_{i\in I}Hom_{X_i-{\bf Set}}(F_i^{\ast}\delta,G_i^{\ast}\epsilon)$$ is a bijection, which was the observation above.