Start a depth-first search from each as-yet unlabelled vertex, propagating a fresh component number:
component = 0
for u in V:
    cc[u] = UNLABELLED
for u in V:
    if cc[u] == UNLABELLED:
        visit(u, component)
        component = component + 1

visit(u, label):
    cc[u] = label
    for v in adjacency[u]:
        if cc[v] == UNLABELLED:
            visit(v, label)
A search follows only edges, so every vertex it labels is reachable from its root. Conversely, if a reachable vertex remained unlabelled, a path to it would contain a first edge from a visited vertex to an unvisited one; scanning that edge would visit the latter, a contradiction. Thus each search labels exactly one connected component, proving both directions of the required equivalence.

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