If I understood it right, the Priest has three intentions when offering Mass and one of them is a specific intentions, eg for a deceased person. When the laity go to Mass what intentions do they offer at the Elevation of the Mass? Do they offer the same intention as the Priest does or their own intentions?

I say a prayer for the deceased person the Mass is offered for, and then I go on to mention my petitions.

The most important thing is to adore, praise and worship Jesus.

even when the Priest doesn’t mention the intention?

Do you refer to my personal intentions? The priest doesn’t have to mention them. The Lord hears them.

Also, when a Mass is offered for a deceased person and their name is announced, it does not mean that only that one person benefits. The Mass is infinite and our faith-filled petitions are heard by the Lord.

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