I fear that I might know far too little category theory to formulate this question more precisely, that why it is only a soft question:
Give some mathematical object $A$, one can often assign to $A$ a numerical quantity $F(A) \in \mathbb R$ whose specific value may often shed light on important features of $A$ in return. Basic examples include:
1) The determinant of an endomorphism $f: V \to V$ on a finite-dimensional $\mathbb R$-vectorspace $V$,
2) The trace of an endomorphism $f: V \to V$ on a finite-dimensional $\mathbb R$-vectorspace $V$,
3) The Euler characterstic on a topological space $X$ admitting a finite $CW$-structure.
In each case, the quantity is usually defined at first with the aid of a specific choice. In the first two cases, one picks an arbitrary basis $B$ of $V$ and computes the corresponding quantities for the resulting square matrix. In the third case, one picks a finite $CW$-structure and computes the alternating sum of numbers of cells.
A major part of why these quantities turn out to be so interesting is of course that they are independent of the choices made (which is why they are also called invariants). In the first two cases, one usually learns this by observing that similar matrices have the same determinant and trace. Only later, after getting to know a little bit on dual spaces and exterior algebra, one finds out that there actually exist base-free definitions of both these quantities.
One usually learns about independence of the choice in the third case by being shown an equivalent, $CW$-free definition (with the aid of singular homology).
In all three cases, a numerical quantity of an object is
a) first defined with aid of a choice, then
b) shown to be independent of that choice, and finally,
c) is equivalently defined without the use of a choice.
My (vague) question now goes as follows:
If a numerical quantity of an object is defined using a specific choice and, within a context-dependent realm of possible choices, is a posteriori shown to be independent of that choice, can that same quantity always be defined without that choice ?