isomorphism of graph definition In Douglas West's book of graph theory, this is how isomorphism of graphs is defined. Please note that graphs need not be simple.

An isomorphism from $G$ to $H$ is a bijection $f$ that maps $V(G)$ to $V(H)$
  and $E(G)$ to $E(H)$ such that each edge of $G$ with end points $u$ and $v$ is
  mapped to an edge with endpoints $f(u)$ and $f(v)$.

Though I get the idea what author seems to say, however I feel the definition is ambiguous in sense that $f: G \to H$. Hence  $f(u)$, $f(v)$ does not make any sense.
Query 1 : Please tell me how right am I in judging the above definition.
Query 2 : Here below I propose my own definition of isomorphism which seems to me a little bit more clear. Please tell me if it is right or not.

Graphs $G$ and $H$ are said to be isomorphic iff there exists bijection
  $f,g$ as defined $f : V(G) \to V(H)$ and  $g : E(G) \to E(H)$ such that
  whenever  $g(e) = e'$  then $u$ and $v$ are end points of $e$ iff $f(u)$ and $f(v)$ are
  end points of $e'$.

Edit: The definition of graph in the book is like this: 

A graph $G$ is a triple consisting of a vertex set $V(G)$ and an edge set
  $E(G)$ and a relation that associates with each edge two vertices not
  necessarily distinct called end points.

Query 3: Upon reading Bondy Murthy's graph theory book's definiton, I think that in above graph definiton won't it be precise to use "function" and not "relation"?
 A: An isomorphism is a bijection (either a bijection $f:V(G) \cup E(G) \to V(H) \cup E(H)$ that sends vertices to vertices and edges to edges, or a pair of bijections $f:V(G) \to V(H)$, $g:E(G) \to E(H)$; these are equivalent). This is correct.  There may be more than one isomorphism from $G \to H$. The two graphs are said to be isomorphic if and only if there exists an isomorphism.  You are giving a definition of what it means for two graphs to be isomorphic, and the book is giving the definition of an isomorphism.
Their definition for the "relation" is indeed a bit strange.  The way they word it, it does sound more like a function taking an edge and returning a set of either one or two vertices (depending on whether the edge is a loop or not).  
But really it is better to consider it as a relation between $E(G)$ and $V(G)$.  We call this relation "incidence" and it can go both ways.  In this case the relation is not a function, because each edge (except loops) is related to two vertices, and each vertex may be related to several edges. 
