What does it mean for two morphisms with different sources and targets to be isomorphic? The standard definition of a subobject relies on the following definition: we call two morphisms $f : X \rightarrow A, g : Y \rightarrow A$ with the same target isomorphic if there exists an isomorphism $u : X \rightarrow Y$ such that $f = g \circ u$. In my eyes, in most texts, this definition is asserted almost by fiat, without diving into what this actually means and why this is a reasonable definition. It's not obvious to me that this is a reasonable definition, for several reasons.
One issue with this definition is that it's not obvious to me why you need it at all - why we can't just define subobjects as isomorphism classes of objects that have monomorphisms into the target in question.
But my larger problem is this. It's obvious what the corresponding definition of isomorphism is between morphisms that share the same source, but what about morphisms that share neither a source nor a target? I would expect that the natural definition would be to say that $f : X \rightarrow A, g : Y \rightarrow B$ are isomoprhic if there exist two isomorphisms $u : X \rightarrow Y, v : A \rightarrow B$ such that $f = v^{-1} \circ g \circ u$. The benefit of this definition is that I believe it flows naturally from the preorder on morphisms in which we define $f \preceq g$ iff $f$ factors through $g$, as the equivalence. However, this definition of isomorphism of morphisms doesn't actually appear to say the same thing as the above 
This definition appears - to me - to the notion of isomorphism in the twisted arrow category, also called the category of factorizations: two morphisms are considered however, when specialized to the case $A = B$, this doesn't actually appear to say the same thing as the previous definition, as we have an extra automorphism tacked on to the end.
So is there a way to generalize isomorphism of morphisms that share the same target to isomorphism of all morphisms? If not, why not?
 A: One way to motivate the definition of subobjects is that it works in familiar categories. For instance, in $\mathbf{Set}$, every subobject of a set $S$ has a unique representative which is a subset of $S$. In particular, two distinct subsets of $S$ are never isomorphic, which would fail under either of your other two proposals. In fact, for $\mathbf{Set}$ the notion of "isomorphism classes of objects admitting a mono into $S$" and "isomorphism class of arrows into $S$" are equivalent, since if $T$ and $T'$ are (without loss of generality) subsets of $S$ and there exists a bijection $\varphi: T\to T'$, then we can always extend $\varphi$ to an automorphism of $S$ making the arrows $T\to S$ and $T'\to S$ isomorphic. 
It's the notion of subset that we want to generalize in defining subobjects, for several very good reasons. The collection of subsets of a set has a rich algebraic structure: it's a complete Boolean algebra. Furthermore, functions between sets induce three morphisms of Boolean algebras which encode first-order logic. Most of this story generalizes to an arbitrary locally Cartesian closed category such as a topos, and some of it generalizes to a huge range of categories satisfying weaker properties. Your proposals amount to considering the set of mere cardinalities at most that of $S$, which has no such interesting structure.
