Why sample covariance also divides by n − 1

The sample covariance of paired data $(X_i, Y_i)$ is defined with the same $n - 1$ divisor as the variance:

$\widehat{\operatorname{Cov}} = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar X)(Y_i - \bar Y).$

Prove that $\mathbb{E}\left[\sum_i (X_i - \bar X)(Y_i - \bar Y)\right] = (n-1)\operatorname{Cov}(X, Y)$, so the divisor is right.