Example 4.13.
When \(C = * = \{X\}\) is the terminal category, we get \(C^{\lex} = (\An^{\fin})\catop\), the opposite of the category of finite animae. In particular, the free logos on one generator is the logos
\[\An[X] \quad := \quad \An[\{X\}] \quad = \quad \PSh((\An^{\fin})\catop) \quad = \quad \Fun(\An^{\fin},\An).\]