TITLE: Preserving equality between different mathematical objects
QUESTION [17 upvotes]: I'm taking an 'Intro to Higher Mathematics'-type course right now, were we learn about basic set theory, number theory, algebra, etc. and I had the following thought: 

Say you're trying to solve a problem in some mathematical field $A$,
  you have objects $a$ and $b$ within theory $A$. You realize that
  objects $c$ and $d$ in the field $B$ are similar to objects $a$ and
  $b$, and that your problem is easier to solve in field $B$. Is there a
  way to make $a \equiv c$ and $b \equiv d$ so that you can go back and
  forth between $A$ and $B$, being sure that whatever it is that you did
  in $B$ is also valid in $A$, and viceversa?

Maybe an example would make what I'm trying to say clearer. (I'm still learning the basics, I'm aware that my knowledge is very limited, so excuse me if my choice of examples is not very good.)
Let $A = \{a,b,c,d\}$. Let $R$ be an equivalence relation on $A$, such that $$R=\{(a,a),(b,b),(c,c),(d,d),(a,b),(b,a),(a,c),(c,a),(b,c),(c,b)\}.$$ 
Now, let's say I want to make this into a directed graph, so let the set of vertices be all the elements of $A$ so that $V=A$ and the set of arcs be the equivalence relation on $A$ such that $E=R$. We have the following graph:

Now, let its adjacency matrix be
$$ M_{ij}= \left( \begin{array}{cccc}
1&1&1&0\\
1&1&1&0\\
1&1&1&0\\
0&0&0&1\\
\end{array} \right), $$ (where $a=1,b=2,c=3,d=4$ for $i,j$).
As we can see, we now have three different ways to approach the same object; now, suppose I want to do something with the initial relation, maybe solve some problem or exercise, my question is: up to what point can I use theorems or results in Graph Theory or Linear Algebra/Matrix Theory to help me solve the problem? 
I know that perhaps my example is very shoddy, but my question is more in general: given some equivalence between some mathematical objects in different theories, is there a way to systematize all the things that I can and cannot do in one theory that preserve the equivalence between the objects I'm using?
E.g, in my previous example, say I operate exclusively on the adjacency matrix of the graph to try to solve my problem; maybe there's some theorem or proposition that makes my problem easier in Linear Algebra than in Set Theory, how can I prove that whatever result I get on the matrix is equivalent to the result I would get working on the sets alone?
To try to summarize, what I'm asking is:

How can I work on some problem from one branch of mathematics to another without losing the properties of the objects I'm working with, so that my results are valid. 
Is there some way to generalize the idea of equivalence between theories.

REPLY [5 votes]: I would definitely say that Category Theory was invented to explore exactly the sort of thing you're asking about. The tricky part is that if the objects in question are similar but not identical (unlike in your example, where you have literally given different representations of the exact same object), then the analog is not perfect, and some things you know about objects $a$ and $b$ may not have analogs or even be meaningful for $c$ and $d$. (In category theory, one deals with forgetful functors between categories, which capture this concept in a general way).