Local smallness of Lawvere theories

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 ?