Let $\mathcal{C}$ be a category. Suppose $\mathcal{C}$ contains a terminal object, which I will denote by $\boldsymbol{1}$. Then for any object $B$ in $\mathcal{C}$, a *global element* of $B$ is a morphism $\boldsymbol{1}\longrightarrow B$. (In many concrete categories, the terminal object is a singleton set, so this definition picks out something close to what we would normally call an "element" of the object $B$.) Given another object $A$ in $\mathcal{C}$, we can say that a morphism $f:A\longrightarrow B$ is *constant* if there is some global element $\epsilon:\boldsymbol{1}\longrightarrow B$ such that $f = \epsilon \circ \alpha$, where $\alpha:A\longrightarrow \boldsymbol{1}$ is the (unique) morphism from $A$ to the terminal object $\boldsymbol{1}$. 


So far, these definitions are totally standard and well-known. But they require the existence of a terminal object. So my questions are the following:



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> Suppose $\mathcal{C}$ is a category which does *not* necessarily have a terminal object. Is there a suitable definition of "global elements" for the objects of $\mathcal{C}$? Is there a suitable definition of "constant morphisms"?
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Here, by a "suitable" definition, I mean a definition which reduces to the definitions I gave above in the case when $\mathcal{C}$ has a terminal object. Also a suitable definition of "constant morphism" should have the following property: *composing any morphism with a constant morphism yields a constant*. (Note that the definition via terminal objects has this property.)



Global elements in categories with no terminal object?
Yes. Instead of working in $C$, you can work in presheaves $[C^{op}, \text{Set}]$ on $C$ using the Yoneda embedding. There is always a terminal presheaf given by sending every object $c \in C$ to $1 \in \text{Set}$ (whether or not it's representable by an object in $C$), and so you can make the following definitions using it.



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> **Definition:** A global element of $c \in C$ is a natural transformation $1 \to \text{Hom}(-, c)$ of presheaves.
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> **Definition:** A constant morphism $f : c \to d$ is a morphism such that the induced morphism $\text{Hom}(- , f) : \text{Hom}(-, c) \to \text{Hom}(-, d)$ factors through the terminal presheaf.
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Because the Yoneda embedding is fully faithful and preserves all limits that exist in $C$, these definitions are guaranteed to reproduce the usual definitions if $C$ does in fact have a terminal object.


Unwinding these definitions, we get the following.



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> **Definition:** A global element of $c$ is a choice, for each object $c' \in C$, of a morphism $f\_{c'} : c' \to c$ such that, for every morphism $g : c' \to c''$, we have $f\_{c''} g = f\_{c'}$.
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> **Definition (edited, 7/6/16):** A constant morphism $f : c \to d$ is a morphism such that there is a choice $f\_{d'} : d' \to d$ of global element of $d$ in the above sense such that, for every morphism $g : c' \to c$, we have $f g = f\_{c'}$.
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