Let $mathcal Z$ be a small category and $mathcal C$ any category. We then consider the category $text{Fun}(mathcal Z, mathcal C)$ the category of functors from $mathcal Z$ to $mathcal C$ and the functor $Gamma: mathcal C to text{Fun}(mathcal Z, mathcal C)$ such that

$$

Gamma(C):

begin{pmatrix}

mathcal Z &longrightarrow &mathcal C\

Z & longmapsto & C\

f & longmapsto & 1_C

end{pmatrix}

$$

where $f: Z to Z’$ and $1_C$ is the identity morphism on $C$.

I am trying to show that $Gamma$ has a left adjoint if and only if the colimit of every functor $F: mathcal Z to mathcal C$. The “if” part is not too hard but I have some difficulties to show the “only if” part.

Here is my idea: Suppose that there is $Omega : text{Fun}(mathcal Z, mathcal C) to mathcal C$ such that

$$theta_{F, C}: text{Hom}_mathcal C(Omega(F), C)cong text{Nat}(F, Gamma(C)).$$

For each $alpha in text{Nat}(F, Gamma(C))$ we can associate a cocone, i.e. for $f: Z to Z’$ the following diagram commutes

$$

begin{array}{ccc} F(Z)&xrightarrow{F(f)} &F(Z’)\ searrow&&swarrow\

&Gamma(C)(Z) = C = Gamma(C)(Z’)& end{array}

$$

where the arrows $F(Z) to C$ and $F(Z’) to C$ are given by $alpha_Z$ and $alpha_{Z’}$ respectively. Because of the above isomorphism, we can associate to $alpha$ a unique morphism $theta^{-1}_{F, C}(alpha) =u: Omega(F) to C$ so $Omega$ is a good candidate to be the colimit of $F$. We just have to find a family of morphism $mu_Z: F(Z) to Omega(F)$ such that $alpha_Z = u circ mu_Z$. My guess is that this family of morphism is given by the unit of the adjunction $eta_F in text{Nat}(F, Gamma(Omega(F)))$ but I am not able to show that

$$alpha_Z = u circ (eta_F)_Z = theta^{-1}_{F, C}(alpha) circ (theta_{F, Omega(F)}(1_{Omega(F)}))_Z$$

where the last equality comes from the expression of $eta_F$ in term of $theta_{F, Omega(F)}$. Does it seem right ? Does anyone know how to conclude ?