  The Fundamental Morphism Theorem is a categorical version of the First
Noether Isomorphism Theorem for categories that do not have kernels or
cokernels. We consider two categories of graphs. Both categories will admit
graphs with multiple edges and loops, and are distinguished by allowing two
different types of homomorphisms, standard graph homomorphisms and more general
graph homomorphisms where the contraction of an edge is allowed. We establish
the Fundamental Morphism Theorem in these two categories of graphs. We then use
the result to provide an equivalent reformulation of the vertex and edge
reconstruction conjectures. This reformulation shows that reconstructability is
equivalent to the existence of a graph homomorphism satisfying an equation.
