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Let $*$\ $\textbf C$ denote the category of pointed objects in $\textbf C$ and sub$\textbf C$ denote the full subcategory of the category of morphisms of $\textbf C$ whose objects are the monomorphisms of $\textbf C$. Then there is a fully faithful inclusion $$ *\text{\ } \textbf C \rightarrow \text{sub} \textbf C$$ if $\textbf C$ is cocomplete does this inclusion always have a left adjoint? How do we compute it? If $\textbf C = \textbf{Top}$ the left adjoint is the quotient space functor... Is what my intuition tells me anyway.... is this right?

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Yes, the inclusion functor does have a left adjoint, which sends a monomorphism $X \hookrightarrow Y$ to its pushout along the unique map $X \to *$.

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