# Finding left/right adjoints of a forgetful functor.

Let $$\mathcal{C}$$ be a category with finite coproducts. Fix some object $$A$$ in $$\mathcal{C}$$. We have a forgetful functor $$U: A/\mathcal{C} \to \mathcal{C}$$ which sends $$f: A \to X$$ to $$X$$ and morphisms to themselves. Does this functor have left and right adjoints?

It does not have a right adjoint, because if it had, then it must preserve colimits, in particular the empty coproduct (= initial object) which is just the identity on $$A$$. In general $$A$$ will not be the initial object in $$\mathcal{C}$$, so in general $$U$$ does not have a right adjoint.

The left adjoint is more difficult. Since it is forgetful, I am guessing it has some free functor as left adjoint, but cannot find a meaningful way to assign a morphism with domain $$A$$ to an object $$C$$ in $$\mathcal{C}$$.

Sometimes it helps to unpack the definition. In this case, for the forgetful functor $$A/\mathcal{C} \to \mathcal{C}$$ to have a left adjoint, you'd need to assign to each object $$X$$ an object $$FX \in \mathcal{C}$$ and a morphism $$f_X : A \to FX$$ such that $$\mathrm{Hom}_{A/\mathcal{C}}(A \xrightarrow{f_X} FX, A \xrightarrow{g} Y) \cong \mathrm{Hom}_{\mathcal{C}}(X, Y)$$ naturally in $$X \in \mathcal{C}$$ and all $$A \xrightarrow{g} Y \in A/\mathcal{C}$$.
Define $$FX = A+X$$ and $$f_X = \iota_A : A \to A+X$$, and note that given $$g : A \to Y$$, morphisms $$h : A+X \to Y$$ such that $$h \circ i_A = g$$ correspond naturally with morphisms $$X \to Y$$.