How to adjoin a terminal object to a category such that each constant morphism factors through it

Let $\mathbf{C}$ be any category. Is there a way to adjoin a terminal object $\ast$ to $\mathbf{C}$ such that each constant morphism factors through any terminal object, i.e., for each constant $f:A\rightarrow B$, $f$ factors as $f:A\rightarrow B = g\circ t_A: A\rightarrow \ast \rightarrow B$ for some $g$

A morphism $f:A\rightarrow B$ is constant if and only if for each $r,s:C\rightarrow A$, we have $f\circ r =f\circ s$.

Obviously this is not always the case, for easy counterexamples you may consider pre-ordered classes with a maximum element, viewed as categories.

I have tried the following approach: Let $\mathbf{C}$ be any category and define $\mathbf{D}$ to be the category with objects $Ob(\mathbf{C})\cup \{\ast\}$ and with morphisms, the morphisms of $\mathbf{C}$ together with $$\{t_D^\ast:D\rightarrow \ast\ |\ D\in Ob(\mathbf{D}) \}\ \ \mathrm{and}$$ $$\{T_f:T\rightarrow B\ |\ \forall\mathrm{\ constant}\ f:A\rightarrow B \mathrm{\ in\ \mathbf{C}}\ \mathrm{and\ }\forall\ \mathrm{terminal\ objects\ } T \mathrm{\ in}\ \mathbf{D} :T_f\circ t_A^T=f \},$$ where $t_A^T$ is the unique morphism from $A$ to $T$ where $T$ is a terminal object in $\mathbf{D}$, i.e., including $\ast$.

Since the morphisms between any two objects, $A$ and $B$ is a set, there can only be set many morphisms from $\ast$ or $T$ to $B$ for any $\mathbf{D}$-object $B$.

My problem arises in definining the composition and checking that this is in fact associative. Clearly $t_\ast = id_\ast$ and in order for any terminal object $T$ to remain terminal, $t_T$ must be an isomorphism. Also, $T_f \circ t_A^T=f$, but how could I define $g\circ T_f$ for some $g:B\rightarrow C$?

Is there a way to describe all the morphisms of $\mathbf{D}$ explicitly, or is there a better approach to generate a smallest category $\mathbf{D}$ that satisfies these properties and for which $\mathbf{C}$ is a subcategory?

• It's not really obvious to me, but how would you define such a forgetful functor? Did you perhaps mean, a forgetful functor from $\mathbf{A}$ to $\mathbf{Cat}$? – sqtrat Apr 19 '16 at 10:54
• Oups sorry, I meant $A\to Cat$ (the inclusion functor). I guess the question is whether the adjoint should be a morphism of 2-categories or something like that. That would depend on the properties you want for your construction. – Captain Lama Apr 19 '16 at 10:57
• Wouldn't that just amount to constructing such a category anyway? – sqtrat Apr 19 '16 at 11:51
• Sure, it's not a solution, you still have to find a construction. It's just a way to state what property this construction should satisfy exactly if it exists (it gives a rigourous meaning to the word "minimal" in your question). – Captain Lama Apr 19 '16 at 12:15
• Yes, of course thank you. – sqtrat Apr 19 '16 at 12:15