In the library you will find a book on summoning, which will tell you which ingredients you need for the first summoning test, most of which are on the summoning level of the academy.
<span class='standouttext'>Spoiler : </span><span class='spoiler'>The Feynur potion is in the laboratory (west side, south room), you trade the imp Nixfix for the Kohar root with the quetzalcellona mushroom you can find in the maintenance tunnels (to the far west of Nixfix and Tax), the spider leg is in the specimen room (east side, north room), and it sounds like you already have the wisdom rune.</span>(highlight the text to read)
The proof you need that you know the tactics of opponents is a book, which is for the second summoning tests. Go talk to the librarian about it and follow that lead.
In detail, <span class='standouttext'>Spoiler : </span><span class='spoiler'> after visiting the library and talking to Antopar, you will find the book was lent to Belehram who gave it to her son Dolehar the drug addict. You should only need to talk to Dotharhar, though, who is in or near the component clerk's room. He tells you an imp may have the book. After that Tax should give you the key to the room his uncle Nixfix locked so you can enter that part of the maintenance tunnel to get the book.</span>
