Is there a category in which, between any two objects, there is a unique morphism? I am interested in knowing information about such a category (if it is well-defined). 
Does there exist a category $\mathcal{C}$, in which there is a unique morphism between any two objects in it?
 A: Yes;


*

*A category in which there is at most one arrow between any two objects is just a poset.

*A category in which there is precisely one arrow between any two objects is just a boolean.
I work up to equivalence, of course.
Edit. Here's a simple way of getting a boolean $\mathrm{isInhab}(X)$ from a set $X$, which expresses the proposition "$X$ is inhabited."


*

*The object set of $\mathrm{isInhab}(X)$ is just $X$.

*The arrow set of $\mathrm{isInhab}(X)$ is $X \times X$.

*The codomain of $(y,x)$ is $y$ and the domain of $(y,x)$ is $x$.

*We define $\mathrm{id}_x = (x,x)$ for each $x \in X$.

*We compose ordered pairs by "cutting" the middle value: $$(z,y) \circ (y,x) = (z,x).$$


Up to equivalence, $\mathrm{isInhab}(X)$ will be "true" if $X$ is inhabited, and "false" if $X$ is the empty set.
A: A category $\mathcal{C}$ in which every hom-set has exactly one element is necessarily a groupoid since the unique morphism $p:A\to B$ and the unique morphism $q:B\to A$ must compose to the morphism $\mathrm{id}_A:A\to A$ and $\mathrm{id}_B:B\to B$.
Moreover, $\mathcal{C}$ is a connected groupoid since every two objects of the category have a morphism between them.
Lastly, $\mathcal{C}$ is a thin category, since every hom-set has at most one element.
So, we could call such a category a thin, connected groupoid.

Such a category is completely determined by its objects, since the requirements for the morphisms are so strict. Any example of such a category is just going to be a set (or class) of objects, to which the necessary unique-morphism-between-any-objects have been added.
The simplest example, then, is the empty category $\mathbf{0}$ (no objects, no morphisms), and the next simplest example after that is the terminal category $\mathbf{1}$ (one object $\ast$, and only $\mathrm{id}_{\ast}:\ast\to\ast$).
A: Given a continuous flow f on a locally compact Hausdorff space X, call a
subset S of X an f-invariant set if S is the union of orbits of the flow f. Call S f-isolated if, with respect to the partial order defined by
inclusion, S is the largest f-invariant set contained in some compact
neighborhood N of S; e.g., the critical elements of a Morse-Smale vector
field on a manifold are isolated relative to the flow generated
by the vector field. The Conley index of an f-isolated invariant set is a
1-connected groupoid (i.e., there is exactly one morphism from A to B for
any ordered pair (A,B) of objects of the groupoid) that is a small
subcategory of the homotopy category of pointed compact Hausdorff spaces.
Conley calls such groupoids connected simple systems. The Conley index is
particularly useful in finding bifurcation orbits, e.g., heteroclinic
orbits between two critical elements. See

*

*H.L. Kurland, The Morse index of an isolated invariant set is a
connected simple system, J.~Differential Equations 42 (1981), 234--259.

*C.C. Conley, Isolated Invariant Sets and the Morse Index, CBMS
Conference Proceedings, Amer. Math. Soc., Providence, R.I., 1978.

