  We follow the same technics we used before in \cite{AZ} of extending knot
Floer homology to embedded graphs in a 3-manifold, by using the Kauffman
topological invariant of embedded graphs by associating family of links and
knots to a such graph by using some local replacements at each vertex in the
graph. This new concept of Graph Floer homology constructed to be the sum of
the knot Floer homologies of all the links and knots associated to this graph
and the Euler characteristic is the sum of all the Alexander polynomials of
links in the family. We constructed three pre-additive categories one for the
graph under the cobordism and the other one is constructed in \cite{AMM} the
last one is a category of Floer homologies for graph defined by Kauffman. Then
we trying to study the functoriality of graphs category and their graph
homologies in two ways, under cobordism and under branched cover, then we try
to find the compatibility between them by the idea of Hilden and Little
\cite{HL} by giving a notion of equivalence relation of branched coverings
obtained by using cobordisms, and hence define a a functor from the graph
Floer-Kauffman Homology category to the graph Khovanov-Kauffman Homology
category.
