Quotient objects in category theory

Question

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?

Answer 1

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