A little correction: an isomorphism is a homomorphism which is bijective and has a homomorphism as its inverse. For many structures the inverse being a homomorphism automatically happens as a logical consequence of the function being a bijective homomorphism, but this isn't the case for all structures. The most prominent examples are topological spaces: a bijection between topological spaces can be continuous (i.e. a homomorphism of topological spaces) but can have an inverse which is not continuous.

The classic example is a function which maps an interval to a circle by applying cos and sin to get the coordinates; this is continuous as small changes in the interval result in a small change in the position on the circumference, but its inverse is not continuous because there will be a point on the circle where the nearby points on one side come from the start of the interval but the nearby points on the other side come from the end of the interval.