A homomorphism can be from a bigger to a smaller graph. Here’s a concrete example:
a
/ \
/ \ 0
b c / \
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d-----e 1-----2
G H
The map $h:V(G)\to V(H)$ given by $h(a)=h(d)=0$, $h(b)=h(c)=1$, and $h(e)=2$ is a graph homomorphism: it sends edges $ab,db$, and $ac$ to $01$, $ce$ to $12$, and $de$ to $02$. Clearly, however, $G$ and $H$ are not isomorphic, since they don’t even have the same number of vertices.
Notice that in general if $h:G\to V$ is a graph isomorphism, then $h(u)h(v)$ is an edge of $H$ if and only if $uv$ is an edge of $G$. If $h$ is merely a graph homomorphism, however, $h(u)h(v)$ can be an edge of $H$ even if $uv$ isn’t an edge of $G$. Thus, if $K$ is the graph shown below, the map that takes $a,b,c,d$, and $e$ to $0,1,2,3$, and $4$, respectively, is a homomorphism but not an isomorphism.
0
/ \
/ \
1-----2
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3-----4
K