Local smallness of Lawvere theories Reading this blog post, I'm trying to care about foundational matters.
To summarize the first part of the article, living in a univers $\mathcal V$ of sets, one defines a Lawvere theory as follow : given a locally small category $\mathbf C$ with a faithful functor $U \colon \mathbf C \to \mathbf{Sets}$, the Lawvere theory $T$ of $\mathbf C$ is the full subcategory of $[\mathbf C, \mathbf{Sets}]$ with objects the finites powers of $U$ : $1,U,U^2,\dots$
For algebraic examples as $\mathbf C = \mathbf{Grps}, \mathbf{Rings}$, etc., one finds $T$ equivalent to  $\mathbf C$, which makes me thinks that we can ensure the local smallness of $T$. But how is that since $[\mathbf C, \mathbf{Sets}]$ is not a priori locally small ? Or is $T$ locally small in the algebraic cases because of the left adjoint of $U$ which makes it representable ?
 A: If $\mathcal{C}$ is locally small and each $U^n : \mathcal{C} \to \mathbf{Set}$ is representable (e.g. because $U : \mathcal{C} \to \mathbf{Set}$ has a left adjoint), then the collection of natural transformations $U^n \Rightarrow U^m$ is a small set (by the Yoneda lemma).
In the case where $\mathcal{C}$ is a locally presentable category and $U : \mathcal{C} \to \mathbf{Set}$ is accessible, $U$ will have a left adjoint. This explains the examples where $\mathcal{C}$ is $\mathbf{Grp}$, $\mathbf{Ring}$, etc. A more interesting example is the case where $\mathcal{C} = \mathbf{KHaus}$ is the category of compact Hausdorff spaces; in that case $\mathcal{C}$ is not locally presentable, but $U$ still has a left adjoint. 
It can also happen that the collection of natural transformations $U^n \Rightarrow U^m$ is a small set when $U$ is not representable. For example, when $\mathcal{C} = \mathbf{Fld}$ is the category of fields. This is because $\mathbf{Fld}$ is an accessible category and each $U^n$ is accessible: all the $U^n$ are determined up to unique isomorphism by their action on the full subcategory of finitely-generated fields, and so too are natural transformations $U^n \Rightarrow U^m$, so we may as well replace $\mathbf{Fld}$ with that essentially small full subcategory.
