Construction 4.9.
Let \(C\) be a small category. We define the free logos on \(C\) as
\[\An[C] \quad := \quad \PSh(C^{\lex}),\]
where \(C^{\lex} \subseteq \PSh(C\catop)\catop\) is defined as the smallest subcategory containing \(C\) which is closed under finite limits. The category \(\An[C]\) comes equipped with a canonical inclusion
\[C \hookrightarrow C^{\lex} \hookrightarrow \PSh(C^{\lex}) = \An[C].\]