### A Question on Left Adjoints

#### Posted by John Baez

*guest post by Jade Master*

I’m interested in internalizing the “free category on a reflexive graph” construction.

We can define reflexive graphs internal to any category $C$, and categories internal to $C$ whenever $C$ has finite limits. Suppose $C$ has finite limits; let $\mathsf{RGph}(C)$ be the category of reflexive graphs internal to $C$, and let $\mathsf{Cat}(C)$ be the category of categories internal to $C$. There’s a forgetful functor

$U \colon \mathsf{Cat}(C) \to \mathsf{RGraph}(C)$

When does this have a left adjoint?

I’m hoping it does whenever $C$ is the category of algebras of a Lawvere theory in $\mathsf{Set}$, but I wouldn’t be surprised if it were true more generally.

Also, I’d really like references to results that answer my question!

Posted at May 28, 2019 6:54 AM UTC
## Re: A Question on Left Adjoints

This functor preserves all limits that exist, so it has a left adjoint whenever the adjoint functor theorem applies. For instance, this is the case when $C$ (hence also $RGraph(C)$ and $Cat(C)$) is locally presentable. This includes the category of algebras for any accessible monad (such as a finitary monad, i.e. Lawvere theory) on a locally presentable category (such as $Set$).