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We prove that a smooth convex body of diameter $$\delta < \frac{\pi }{2}$$ on the d-dimensional unit sphere $$S^d$$ is of constant diameter $$\delta $$ if and only if it is of constant width $$\delta $$. We also show this equivalence for all convex bodies on $$S^2$$. Since, as shown earlier, the equivalence on $$S^d$$ is true for every $$\delta \ge \frac{\pi }{2}$$, the question whether spherical bodies of constant diameter and constant width on $$S^d$$ coincide remains open for non-smooth bodies on $$S^d$$, where $$d\ge 3$$.

When is a spherical body of constant diameter of constant width?