A finger is a component of a hand, a hand is the X of a finger

There is no general purpose way of going from the formulation "A is a part of B" to "B is a (blank) of A".

Even in semantics, this "B is a (blank) of A" is described as "B has an A".

This can be seen in the discussions of the semantic-relationship between the names of objects, using the terms meronym and holonym.

Meronymy and holonymy cover the cases like

• X is a part of Y
• X has a Y

by referring the relationship between the names of X and Y:

"X" is a meronym of "Y" if Xs are parts of Ys

"X" is a holonym of "Y" if Ys are parts of Xs (Xs have Ys)

Strictly speaking, this relationship is about the words themselves and how they describe the relationships between their corresponding objects.

So:
The word "finger" is a meronym of the word "hand" if fingers are parts of hands.
The word "hand" is a holonym of the word "finger" if hands have fingers.

But my point is, even in semantics, there is no other way to describe the relationship between the objects rather than the form "A has a B".

Of course there are specific cases where there are words. *Parent-Child" is one of them. There are others: Part-Whole; Source-Product, as examples.

I strongly question your categories here as not logical. E.g.:

Milk is a product of cows, cows are a source of leather; a source-product relationship.

Milk and leather are both products, with no relationship other than that. Alligator leather has nothing in common with milk, save that it is an animal product. Eggs are animal products; they have nothing in common with leather, except for being animal products. Your category is illogical.

A finger is a component of a hand, a hand is the __ of a finger; a __-component relationship.

The finger is indeed a part or component of the hand; the hand is the whole, which contains all the parts.