and where is the rest of the transaction (the summary ?) ... in mempool ?
Full nodes still have both parts, the transaction and the signature. Same as before they dont store them in mempool once confirmed, but on disk.
when you need to emit transaction, you must have the complet transmission in the block.
so, SW help during the building block (before) ... but not when the block must be accepted.
so, the question is ... how many transaction (at the most) can be here in a block ?
2200 transactions in 10 min ?
3900 transactions in 10 min ?
SW would basically introduce a new type of transaction. Like we now have type 1 and pay2script hash (3), we get e.g. 4 (because 2 and 5 are taken) for segwit transactions. These new type of transactions would be significantly smaller because they store the signature outside of the part that counts. The signature part would also be only counted as 25% of its actual size. Once SW is introduced properly (soft fork first, hard fork later) it would be easy to introduce new transaction types, by just adding a new prefix with a soft fork. This could e.g. make pay to script hash significantly smaller, because you could define the script as "outside" of the transaction and no longer counting towards the blocksize limit.
It essentially reduces a TX (or the part that is counted towards the blocklimit) to "move coins from A to B".