Let $F\colon \mathcal{C}\rightarrow\mathcal{D}$ a functor. Then, one obtains a functor $\tilde F\colon \textbf{Set}^\mathcal{D}\rightarrow \textbf{Set}^\mathcal{C}$ (composing with $F$). I want to show that $\tilde F$ has a left adjoint by using Freyd's Adjoint Functor Theorem. I have checked all the assumptions but the existence of a weakly initial set of objects in the comma category $(G\downarrow\tilde F)$, where $G$ is any functor $\mathcal{C}\rightarrow \textbf{Set}$. That's where I have the problem. I think I have to choose a set of the form $\{ ( \mathcal{D}( FA, -),\eta^{A})\ \mid A\in |\mathcal{C}| \}$, but I don't know how to choose the natural transformations $\eta^A$ (or maybe I should take coproduts of representable functors?). Can anyone give me a hint?

  • $\begingroup$ Presumably, you're assuming at least $\mathcal C$ (and probably also $\mathcal D$) is small, or else $\{(\mathcal D(FA,-),\eta^A)\mid A\in|\mathcal C|\}$ isn't a set because $|\mathcal C|$ isn't a set. $\endgroup$ – Derek Elkins Feb 21 '18 at 19:50
  • $\begingroup$ Also, for each $G$ you need to choose a weakly initial family, so presumably that family would depend on $G$ in some way. (Of course, in this case since we can directly describe the left adjoint, we could just write down the initial object in the comma category... You could cheat and work backwards to some plausible weakly initial family...) $\endgroup$ – Derek Elkins Feb 21 '18 at 19:55
  • $\begingroup$ Yes - both categories are assumed to be small. $\endgroup$ – J. Karen Feb 21 '18 at 23:32

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