In a set of notes, I came across the following few lines involving the covariant derivative, and just wanted to make sure I understood the notation correctly:

Let $\lbrace F_{1},F_{2},F_{3},F_{4}\rbrace$ be a null tetrad. Then from Newman Penrose formalism we have an equation

$\nabla_{F_{4}}F_{3}=(\alpha-\overline{\beta})F_{3}+\overline{\mu}F_{1}-\rho F_{2}$

Then we have

$F_{4}\hskip .5pt ^{l}\nabla_{l}F_{3}\hskip .5pt^{k}=(\alpha-\overline{\beta})F_{3}\hskip .5pt ^{k}+\overline{\mu}F_{1}\hskip .5pt ^{k}-\rho F_{2}\hskip .5pt ^{k}$

So, do I understand correctly that in the second equation, we simply take the components of the original Newman Penrose equation?